Frequency-temperature superposition principle

Upon substitution of (9.82) into the moduli (9.68), we find the frequency-temperature superposition principle such that a modulus-frequency curve at any temperature T can be superimposed onto a single curve at the reference temperature 7b, if it is vertically and horizontally shifted properly. Such construction of the master curve is described by the equation... 

The time-temperature superpositioning principle was applied f to the maximum in dielectric loss factors measured on poly(vinyl acetate). Data collected at different temperatures were shifted to match at Tg = 28 C. The shift factors for the frequency (in hertz) at the maximum were found to obey the WLF equation in the following form log co + 6.9 = [ 19.6(T -28)]/[42 (T - 28)]. Estimate the fractional free volume at Tg and a. for the free volume from these data. Recalling from Chap. 3 that the loss factor for the mechanical properties occurs at cor = 1, estimate the relaxation time for poly(vinyl acetate) at 40 and 28.5 C. 

Whatever the selected method (static, monotonous, or dynamic), it gives access to a limited range of timescales. For example it is almost impossible to perform static experiments in times less than 1 s, or dynamic tests at frequencies lower than 10 1 Hz, or tensile tests at strain rates higher than 103 s-1. These timescales are, however, indirectly accessible because the polymers generally obey a time-temperature superposition principle ... 

The peak of the dielectric loss of this process reflects its viscoelastic nature by obeying the time-temperature superposition principle, wherein the peak is shifted to higher temperatures for shorter times (higher frequencies) and vice versa. This process has been described by the Havriliak-Negami empirical formula [106, 108]... 

This is none other than the time-temperature superposition principle. However, the exact shape of the function F(t) is not a generic feature of the theory. A good approximation is provided by the Kohlrausch stretched exponential function. In the frequency domain, Eqs. (23) define the shape of the susceptibility minimum usually observed in the gigahertz range, and an interpolation formula follows ... 

The Time-Temperature Superposition Principle. For viscoelastic materials, the time-temperature superposition principle states that time and temperature are equivalent to the extent that data at one temperature can be superimposed upon data at another temperature by shifting the curves horizontally along the log time or log frequency axis. This is illustrated in Figure 8. While the relaxation modulus is illustrated (Young s modulus determined in the relaxation mode), any modulus or compliance measure may be substituted. 

Almost always the data from the apparatus above is analyzed by using the time-temperature superposition principle to form a master curve over a wide frequency range at a selected reference temperature. The basis for this procedure is that for thermorheologically simple materials the effect of a change in temperature on... 

Assuming that the time-temperature superposition principle holds, estimate the frequency at which the modulus at Tg + 50 is equal to the modulus at Tg + 20. Assume that Tg — 100°C. 

However, for thermorheologically simple materials, that is, for those materials for which the time-temperature superposition principle holds, the mechanical properties data can be shifted parallel to the time or frequency axis. This fact suggests an additional hypothesis that can be very useful in solving some specific thermoviscoelastic problems. According to this hypothesis, the net effect of temperature in the response must be equivalent to a variation in the rates of creep or relaxation of the material. Thus for T > Tq the process occurs at a higher rate than at Tq. 

Demonstration of the time-temperature superposition principle, using oscillatory shear data (G, filled circles and G", open diamonds) on a PVME melt with M — 124000 gmol. The right-hand plot shows the data that were acquired at the six temperatures indicated, with Tg = - 24°C chosen as the reference temperature. All data were shifted empirically on the modulus and frequency scales to superimpose, constructing master curves for G and G" in the left-hand plot. Data and... 

For blends of LDPE with EMA-salts, G is superposable but there is a clear breakdown of the time-temperature superposition principle at high frequencies for G". Furthermore, the frequency range over which G" is superposable decreases with increasing ionomer (EMA-salt) content. In other words, the inability of EMA-salt to Increase G" at high frequencies, as a function of temperature, increases as the content of EMA-salt in the blend is Increased. These results lead to the conclusion that above the crystalline melting temperature of the two components, the breakdown of the time-temperature superposition principle in G" is due solely to the presence of ionic domialns in PE/EMA-salt blends. 

These data can be used to show that time and temperature effects are often coupled for relaxation phenomena (the time/temperature superposition principle [13]). Effects due to an temperature increase can thus also be obtained by an increase of the experimental time scale. Hence, the curves in Figure 5.8 were shifted along the frequency-axis while the curve measured at 18°C was chosen as reference temperature. 

The time-temperature superposition principle, t-T, has been a cornerstone of viscoelastometry. It has been invariably used to determine the viscoelastic properties of materials over the required 10 to 15 decades of reduced frequency, COaj, [Ferry, 1980]. Measuring the rheological properties at several levels of temperature, T, over the experimentally accessible frequency range (usually two to four decades wide), then using the t-T shifting, made it possible to constmct the complete isothermal function. 

There is growing evidence that t-T superposition is not valid even in miscible blends well above the glass transition temperature. For example, Cavaille et al. [1987] reported lack of superposition for the classical miscible blends — PS/PVME. The deviation was particularly evident in the loss tangent vs. frequency plot. Lack of t-T superposition was also observed in PI/PB systems [Roovers and Toporowski, 1992]. By contrast, mixtures of entangled, nearly mono-dispersed blends of poly(ethylene-a/f-propylene) with head-to-head PP were evaluated at constant distance from the glass transition temperature of each system, homopolymer or blend [Gell et al, 1997]. The viscoelastic properties were best described by the double reptation model , viz. Eq 7.82. The data were found to obey the time-temperature superposition principle. 

The most common means to extend the frequency scale is to invoke time-temperature superpositioning (Ferry, 1980). If all motions of a polymer contributing to a particular viscoelastic response are affected the same by temperature, then changes in temperature only alter the overall time scale such a material is thermorheologically simple. Thermorheological simplicity means conformance to the time-temperature superposition principle, whereby lower and higher strain rate data can be obtained from measurements at higher and lower temperatures, respectively. 

The observed glass transition temperatures (T ) of several thin polymer films on surface acoustic wave (SAW) devices are 50-60 °C higher than the results reported using other methods such as DSC. The increase in the onset of T is the result of interaction of the high frequency SWJ with the polymer film, consistent with the time-temperature superposition principle. The Tj were identified as localized minima in the frequency curves, or by changes in the slope of the curves, as the coated sensors were heated between 35-110 °C. Potential applications of SAWs for the characterization of polymer materials and the Implications of these findings for the interpretation of SAW data are discussed. 

By application of the time-temperature superposition principle, a decade of frequency can be shown to correspond to a 6 or TC shift in 7. Noting that the normal acoustical range goes from 20 to 20,000 Hz, or three decades, it can be seen that the equivalent temperature range is 18-20°C. We then conclude that a properly chosen homopolymer can Just damp all acoustical frequencies at a single use temperature. 

Fig. 6.12 Illustration of (a) the storage modulus, the loss modulus and the loss factor as a function of frequency across the glass transition temperature of amorphous polymers (b) the loss factor as a function of temperature according to the time-temperature superposition principle. Below the a peak for glass transition, there are secondary relaxation peaks...

By the application of the time-temperature superposition principle described in Sect. 2.2.9, moduli were obtained over a frequency range up to seven decades. In Fig. 27, master curves of 9.9 wt% PVC/DOP are shown after ageing over 6 days at different temperatures. From this figure the following can be concluded ... 

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